6 problems

Critical-frequency stability conjecture. For each $p>6$, there exists a critical value $\omega^*(p)$ such that the waves $\phi_{\omega,p}$ are unstable for $\omega>\omega^*(p)$ and…

Let $v$ be a solitary-wave solution of the Benjamin equation, with spatial variable $x\in\mathbb R$. The known asymptotic result gives algebraic decay of the profile at spatial inf…

Consider the Benjamin solitary-wave equation … Here $v$ is a solitary-wave profile, $c$ is its velocity, and $\alpha,\beta$ are the Benjamin-equation parameters. Solitary-wave exis…

Eventual-separation conjecture. These clusters should eventually separate into distinct solitary waves.

Chen–Xiao's conjecture. The initial value problem $$ is locally well-posed for $s=-3/4$.

Consider the Benjamin equation … with initial data $u(x,0)=u_0(x)$ on $\mathbb{R}$, where $\mathcal{H}$ is the one-dimensional Hilbert transform and $H^s(\mathbb{R})$ denotes the S…